| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s + 4·13-s − 2·15-s + 17-s + 6·19-s + 2·21-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s − 4·35-s + 2·37-s − 4·39-s − 2·41-s + 2·43-s + 2·45-s − 3·49-s − 51-s − 6·53-s − 6·57-s + 4·59-s − 4·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s + 0.242·17-s + 1.37·19-s + 0.436·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.640·39-s − 0.312·41-s + 0.304·43-s + 0.298·45-s − 3/7·49-s − 0.140·51-s − 0.824·53-s − 0.794·57-s + 0.520·59-s − 0.512·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.076864578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.076864578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.55585250033556, −13.31078813127337, −12.74737706676203, −12.59340569554407, −11.56846986829376, −11.36822804303716, −10.89292486002524, −10.30221658444814, −9.733684697625014, −9.406444222412124, −9.018181275427776, −8.353714847439164, −7.563101039268094, −7.117057717757797, −6.676573070542424, −5.992080766743383, −5.549401023671200, −5.395565378582684, −4.502572638403094, −3.793971250308304, −3.223758450811457, −2.784869587002109, −1.668173668561168, −1.391946857032766, −0.4818390235086302,
0.4818390235086302, 1.391946857032766, 1.668173668561168, 2.784869587002109, 3.223758450811457, 3.793971250308304, 4.502572638403094, 5.395565378582684, 5.549401023671200, 5.992080766743383, 6.676573070542424, 7.117057717757797, 7.563101039268094, 8.353714847439164, 9.018181275427776, 9.406444222412124, 9.733684697625014, 10.30221658444814, 10.89292486002524, 11.36822804303716, 11.56846986829376, 12.59340569554407, 12.74737706676203, 13.31078813127337, 13.55585250033556
